Quadrupole density function

Schwerdtfeger, P., Bast, R., Gerry, M.C.L., Jacob, C.R., Jansen, M., Kelld, V., Mudring, A.V., Sadlej, A.J., Saue, T, Sdhnel, T. and Wagner, F.E. (2005) The quadrupole moment of the 3 /2 nuclear groimd state of Au from electric field gradient relativistic coupled cluster and density functional theory of small molecules and the solid slide. Journal of Chemical Physics, 122,124317-1-124317-9. 

Thierfelder, C., Schwerdtfeger, P. and Saue, T. (2007) Cu and Au Nuclear Quadrupole Moments from Four-Component Relativistic Density Functional Calculations using Exact Long-Range Exchange. Physical Review A, 76, 034502-1-034502-4. 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The spherical harmonic density functions are referred to as multipoles, since the functions with 1 = 0, 1, 2, 3, 4, etc., correspond to components of the charge distribution p r) which give nonzero contributions to the monopole (/ = 0), dipole (/ = 1), quadrupole (/ = 2), octupole (/ = 3), hexadecapole (/ = 4), etc., moments of the atomic charge distribution. 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

During the early years at McGill University, Whitehead s group concentrated on experimental nuclear quadrupole resonance spectroscopy123 and a variety of n- and all-valence electron semiempirical molecular orbital methods.124 His recent interests have included topics as diverse as density functional theory125 and related topics,126 and molecular models of surfactants. 

The quadrupole splitting A Eg is the first Mossbauer parameter that could be predicted with reasonable accuracy by electronic structure calculations. The quadrupole sphtting can be related to the EFG at the Mossbauer nucleus according to equation (12). Since the nuclear quadrupole moment Q is very difficult to determine by experiment alone, the combination of Mossbauer spectroscopy and density functional calculations is currently the most accurate approach to estimate Q. A linear regression of measured quadrupole splittings and calculated electric field gradients has been performed by Blaha et al. for a large series of compounds... 

In Table 1 we summarize their main findings. For the frozen-density calculations they considered two different approaches, one where the solvent-molecule density was kept fixed and one where it was allowed to relax. In the table we have only shown the results for the latter, which according to the authors led to an improved accuracy. The table shows that the dipole and the quadrupole moments are very similar for both approaches, which is to a lesser extent the case for the excitation energies and the static (hyper)polarizabilities. The latter were calculated using time-dependent density-functional theory. In order to understand this discrepancy the authors used also a supermolecule approach with just two solvent molecules. By comparing with results from calculations with the frozen-density and the polarizable-molecule approaches on the same system they concluded that the frozen-density approach was the more accurate one in calculating the responses to electromagnetic fields. 

Zhang, Y., Mao, J.H., Godbout, N., Oldfield, E. Mossbauer quadrupole splittings and electronic structure in heme proteins and model systems A density functional theory investigation. J. Am. Chem. Soc. 2002,124,13921-30. 

An important recent development comes from the recognition that an electronic density function written in terms of a gaussian basis set can be expressed as a finite multicenter expansion.(The centers are not limited to nuclei.) The resulting V(r), based on an expansion through the quadrupole terms, has been found to agree well with that obtained rigorously from the same p(r), with a very marked reduction in the computer time required." ... 

More recently, Cohen and Tantirungrotechai81 explored the performance of several density-functional methods in calculating dipole and quadrupole moments as well as polarizabilities of some smaller molecules. Besides confirming the conclusions above, they, found less clear trends for the quadrupole moments as well as the fact that the hybrid methods yield slightly better agreement with experiment both for the static dipole moment and for the polarizabilities. 

CW = continuous-wave CW-NQR = continuous wave NQR DFT = density functional theory EFG = electric field gradient IR = infrared NMR = nuclear magnetic resonance NQR = nuclear quadrupole resonance OSSE = octahedral site stabilization energy PAC = perturbed angular correlation pulse-FT = pulse-fourier transform TMED = tetramethylethylenediamine. 

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